Absolute Value Functions and Graphs

Absolute Value Functions and

Graphs

Lesson 2-5

Important Terms• Parent function: the simplest function with these

characteristics. The equations of the function in a family resemble each other, and so do the graphs. Offspring of parent functions include translations, stretches, and shrinks.

• Translation: it shifts a graph horizontally, vertically, or both. It results in a graph of the same shape and size but possibly in a different position

• Stretch: a vertical stretch multiplies all y-values by the same factor greater than 1, thereby stretching the graph vertically

• Shrink: a vertical shrink reduces y-values by a factor between 0 and 1, thereby compressing the graph vertically

• Reflection: in the x-axis changes y-values to their opposites. When you change the y-value of a graph to their opposites, the graph reflects across the x-axis (creates a mirror image)

The Family of Absolute Value FunctionsVertical Translation

Parent function Y=|x| Y=f(x)

Translation up k units, k>0 Y=|x|+k Y=f(x)+k

Translation down k units, k<0 Y=|x|-k Y=f(x)-k

Horizontal Translation

Parent Function Y=|x| Y=f(x)

Translation right h units, h>0 Y=|x-h| Y=f(x-h)

Translation left h units, h<0 Y=|x+h| Y=f(x+k)

Combined Translation

(right h units, up k units) Y=|x-h|+k Y=f(x-h)+k

Families of Functions: Absolute Value Functions

Vertical Stretch or Shrink, and Reflection in x-axisParent function Y=|x| Y=f(x)

Reflection in x-axis Y=-|x| Y= -f(x)

Stretch (a>1) Y=a|x| Y=af(x)

Shrink (0<a<1)

Reflection in x-axis Y=-a|x| Y=-af(x)

Combined Translation

Y=a|x-h|+k Y=af(x-h)+k

Absolute Value

An Absolute Value graph is always in a “V” shape.

xy

Given the following function,

If: a > 0, then shift the graph “a” units up

If: a < 0, then shift the graph “a” units down

xy a

Given the following function,

Since a > 0, then shift the

graph “3” units up

3xy

Let’s Graph

3xy

5xy

How will the graph look?

Let’s Graph

5xy

2xy

How will the graph look?

Let’s Graph

2xy

4xy

How will the graph look?

Let’s Graph

4xy

Given the following function,

We get the expression (x - b) and equal it to zero

x - b = 0x = b

If: b > 0, then shift the graph “b” units to the right

If: b < 0, then shift the graph “b” units to the left

x by

Given the following function,

x – 1 = 0

x = 1

Since 1 > 0, then shift

the graph “1” unit right

1xy

Let’s Graph

1xy

6xy

How will the graph look?

Let’s Graph

6xy

3xy

How will the graph look?

Let’s Graph

3xy

7xy

How will the graph look?

Let’s Graph

7xy

Graphing

1 3xy

Recall: Shift “3” units up since 3 > 0then we use the expression x + 1,

and equal it to zerox + 1 = 0

x = -1Since –1 < 0, then we shift

“1” unit to the left

Let’s Graph

1 3xy

3 2xy

How will the graph look?

Let’s Graph

3 2xy

2 4xy

How will the graph look?

Let’s Graph

2 4xy

5 1xy

How will the graph look?

Let’s Graph

5 1xy

Given the following function,

For this equation, c determines

how wide or thin it will be.if: |c|>1, then the graph is closer to the y-axis

if: |c|=1, then the graph remains the same

if: 0<|c|<1, then the graph is further

from the y-axis

if c is a negative number, then the graph

will reflect on the x-axis

xy c

Given the following function,

Since |5| > 0, then the

graph is closer to the y-axis

5 xy

Let’s Graph

5 x

xy

y

4 xy

How will the graph look?

Let’s Graph

4 x

xy

y

1

2xy

How will the graph look?

Let’s Graph

1

2x

xy

y

5

4xy

How will the graph look?

Let’s Graph

5

4x

xy

y

2

3xy

How will the graph look?

Let’s Graph

2

3

x

x

x

y

y

y

Given the following function,

Since 4 > 0, shift the graph “4” units up

x – 1 = 0

x = 1

Since 1 > 0, then shift the graph

“1” unit to the right

Since |5| > 0 shift the graph

closer to the y-axis.

1 45 xy

Let’s Graph

15 4xy

53 2xy

How will the graph look?

Let’s Graph

53 2xy

42 3xy

How will the graph look?

Let’s Graph

42 3xy

31

62xy

How will the graph look?

Let’s Graph

31

62xy

45

24xy

How will the graph look?

Let’s Graph

45

24xy

29

44xy

How will the graph look?

Let’s Graph

29

44xy

52

33xy

How will the graph look?

Let’s Graph

52

33xy

14

53xy

How will the graph look?

Let’s Graph

14

53xy

Congratulations!!You just completed the

transformation of

y x